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Math Help - Algebraic system

  1. #1
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    Algebraic system

    Define on RXR by setting (a,b)(c,d)=(acbd,ad+bc)Show that (RXR,) is an algebraic system
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  2. #2
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    Re: Algebraic system

    What do you mean by algebraic system? A ring? A field?
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  3. #3
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    Re: Algebraic system

    Well I havent learnt abt any algebraic structures except a group. I am wondering what defines an algebraic structure
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  4. #4
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    Re: Algebraic system

    the algebraic system formed by (RxR,⊕) is called a commutative monoid (which is almost like a group, except some elements don't have inverses). it's "almost" an abelian group. in fact, it would be an abelian group if you left out (0,0). so what you can prove:

    ⊕ is associative.
    ⊕ is commutative.
    ⊕ has an identity (what is it?)
    every element except (0,0) has an inverse.

    EDIT: formally, an "algebra" is:

    a vector space V, together with an operation * such that * is bilinear, and distributive over + (vector sums).

    in other words:

    (V,+) is an abelian group
    there is a map FxV → V (usually called scalar multiplication, where F is a field) with (for all a,b in F and u,v in V):

    a(u+v) = au + av
    (a+b)u = au + bu
    a(bu) = (ab)u
    1u = u

    and (V,+,*) is a ring, with a(u*v) = (au)*v = u*(av).

    it turns out that (RxR,+,⊕) is actually an algebra under this definition, as well, if we define the vector addition and scalar multiplication like so:

    (a,b) + (c,d) = (a+c,b+d)
    r(a,b) = (ra,rb). i doubt you are being asked to show this, but it's possible.
    Last edited by Deveno; August 16th 2012 at 04:45 PM.
    Thanks from mathrld
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